State whether the following statements are true or false. Justify your answer.
Point $ P(0,2) $ is the point of intersection of $ y $-axis and perpendicular bisector of line segment joining the points $ A(-1,1) $ and $ B(3,3) $.

Point \( P(0,2) \) is the point of intersection of \( y \)-axis and perpendicular bisector of line segment joining the points \( A(-1,1) \) and \( B(3,3) \).

To do:

We have to find whether the given statement is true or false.

Solution:

Let us assume that the given statement is true.

This implies,

The point \( P(0,2) \) lies on \( y \)-axis.

$P(0,2)$ lies on the perpendicular bisector of $AB$

This implies,

$AP=BP$

$AP=\sqrt{(0+1)^{2}+(2-1)^{2}}$

$=\sqrt{2}$

$BP=\sqrt{(0-3)^{2}+(2-3)^{2}}$

$=\sqrt{9+1}$

$=\sqrt{10}$

$A P ≠ B P$

Therefore,

The point $P$ does not lie on the perpendicular bisector of line segment joining the points \( A(-1,1) \) and \( B(3,3) \).

The given statement is false.